Problem: Define a sequence of complex numbers by $z_1 = 0$ and
\[z_{n + 1} = z_n^2 + i\]for all $n \ge 1.$  In the complex plane, how far from the origin is $z_{111}$?
Answer: The first few terms are
\begin{align*}
z_2 &= 0^2 + i = i, \\
z_3 &= i^2 + i = -1 + i, \\
z_4 &= (-1 + i)^2 + i = -i, \\
z_5 &= (-i)^2 + i = -1 + i.
\end{align*}Since $z_4 = z_2,$ and each term depends only on the previous term, the sequence from here on is periodic, with a period of length 2.  Hence, $|z_{111}| = |z_3| = |-1 + i| = \boxed{\sqrt{2}}.$